Skip to main content
SpringerLink
Log in

Asymptotic Dynamics in Populations Structured by Sensitivity to Global Warming and Habitat Shrinking

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

How to recast effects of habitat shrinking and global warming on evolutionary dynamics into continuous mutation/selection models? Bearing this question in mind, we consider differential equations for structured populations, which include mutations, proliferation and competition for resources. Since mutations are assumed to be small, a parameter ε is introduced to model the average size of phenotypic changes. A well-posedness result is proposed and the asymptotic behavior of the density of individuals is studied in the limit ε→0. In particular, we prove the weak convergence of the density to a sum of Dirac masses and characterize the related concentration points. Moreover, we provide numerical simulations illustrating the theorems and showing an interesting sample of solutions depending on parameters and initial data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. IPCC Fourth Assessment Report: Climate Change 2007 (AR4)

  2. Bairagi, N., Jana, D.: On the stability and Hopf bifurcation of a delay-induced predator-prey system with habitat complexity. Appl. Math. Model. 35, 3255–3267 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barles, G., Mirrahimi, S., Perthame, B.: Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result. Methods Appl. Anal. 16, 321–340 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berestycki, H., Diekman, O., Nagelkerke, K., Zegeling, P.: Can a species face a climate change? Bull. Math. Biol. 71, 399–429 (2008)

    Article  Google Scholar 

  5. Brauer, F., Sanchez, D.A.: Periodic environments and periodic harvesting. Nat. Resour. Model. 16, 233–244 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Calsina, À., Cuadrado, S.: Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics. J. Math. Biol. 48, 135–159 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cantrell, R.S., Cosner, C., Fagan, W.F.: Brucellosis, botflies, and brainworms: the impact of edge habitats on pathogen transmission and species extinction. J. Math. Biol. 42, 95–119 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Champagnat, N., Jabin, P.E.: The evolutionary limit for models of populations interacting competitively with many resources. J. Differ. Equ. 251, 176–195 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Champagnat, N., Ferrière, R., Méléard, S.: Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theor. Popul. Biol. 69, 297–321 (2006)

    Article  MATH  Google Scholar 

  10. Cozzens, M.: Food webs, competition graphs, and habitat formation. Math. Model. Nat. Phenom. 6, 622–638 (2011)

    Article  MathSciNet  Google Scholar 

  11. Delitala, M., Lorenzi, T.: Asymptotic dynamics in continuous structured populations with mutations, competition and mutualism. J. Math. Anal. Appl. 389, 439–451 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Desvillettes, L., Jabin, P.E., Mischler, S., Raoul, G.: On selection dynamics for continuous structured populations. Commun. Math. Sci. 6, 729–747 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Diekmann, O., Jabin, P.-E., Mischler, S., Perthame, B.: The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach. Theor. Popul. Biol. 67, 257–271 (2005)

    Article  MATH  Google Scholar 

  14. Evans, S.N., Ralph, P.L., Schreiber, S.J., Sen, A.: Stochastic population growth in spatially heterogeneous environments. J. Math. Biol. 66, 423–476 (2011)

    Article  MathSciNet  Google Scholar 

  15. Ferriere, R., Guionnet, A., Kurkova, I.: Timescales of population rarity and commonness in random environments. Theor. Popul. Biol. 69, 351–366 (2006)

    Article  MATH  Google Scholar 

  16. Foppen, R., Braak, C.J.F.T., Verboom, J., Reijnen, R.: Dutch sedge warblers Acrocephalus schoenobaenus and West-African rainfall: Empirical data and simulation modelling show low population resilience in fragmented marshlands. Ardea 87, 113–127 (1999)

    Google Scholar 

  17. Foufopoulos, J., Kilpatrick, A.M., Ives, A.R.: Climate change and elevated extinction rates of reptiles from Mediterranean Islands. Am. Nat. 177, 119–129 (2011)

    Article  Google Scholar 

  18. Hanski, I., Gaggiotti, O.E. (eds.): Ecology, Genetics, and Evolution of Metapopulations. Elsevier, Amsterdam (2004)

    Google Scholar 

  19. Henson, S.M., Cushing, J.M.: The effect of periodic habitat fluctuations on a nonlinear insect population model. J. Math. Biol. 36, 201–226 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jabin, P.E., Raoul, G.: On selection dynamics for competitive interactions. J. Math. Biol. 63, 493–517 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kussell, E., Leibler, S.: Phenotypic diversity, population growth, and information in fluctuating environments. Science 309, 2075–2078 (2005)

    Article  Google Scholar 

  22. Lorz, A., Mirrahimi, S., Perthame, B.: Dirac mass dynamics in multidimensional nonlocal parabolic equations. Commun. Partial Differ. Equ. 36, 1071–1098 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Magal, P., Webb, G.F.: Mutation, selection and recombination in a model of phenotype evolution. Discrete Contin. Dyn. Syst. 6, 221–236 (2000)

    MATH  MathSciNet  Google Scholar 

  24. Mirrahimi, S.: Adaptation and migration of a population between patches. Discrete Contin. Dyn. Syst., Ser. B 18, 753–768 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  25. Mirrahimi, S., Perthame, B., Wakano, J.Y.: Evolution of species trait through resource competition. J. Math. Biol. 64, 1189–1223 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Opdam, P., Wascher, D.: Climate change meets habitat fragmentation: linking landscape and biogeographical scale levels in research and conservation. Biol. Conserv. 117, 285–297 (2004)

    Article  Google Scholar 

  27. Pei, Y., Zeng, G., Chen, L.: Species extinction and permanence in a prey-predator model with two-type functional responses and impulsive biological control. Nonlinear Dyn. 52, 71–81 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Perthame, B., Barles, G.: Dirac concentrations in Lotka-Volterra parabolic PDEs. Indiana Univ. Math. J. 57, 3275–3301 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Rifkin, J.: The Third Industrial Revolution: How Lateral Power Is Transforming Energy, the Economy and the World. Palgrave Macmillan, New York (2011)

    Google Scholar 

  30. Rifkin, J.: Talk at Pirelli Sustainability Day. http://sustainabilityday.pirelli.com/sdayPost2012/

  31. Rivoire, O., Leibler, S.: The value of information for populations in varying environments. J. Stat. Phys. 142, 1124–1166 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Rizaner, F.B., Rogovchenko, S.P.: Dynamics of a single species under periodic habitat fluctuations and Allee effect. Nonlinear Anal., Real World Appl. 13, 141–157 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  33. Sagitov, S., Jagers, P., Vatutin, V.: Coalescent approximation for structured populations in a stationary random environment. Theor. Popul. Biol. 78, 192–199 (2010)

    Article  Google Scholar 

  34. Schreiber, S.J., Benaïm, M., Atchadé, K.A.S.: Persistence in fluctuating environments. J. Math. Biol. 62, 655–683 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  35. Weng, P., Zhao, X.-Q.: Spatial dynamics of a nonlocal and delayed population model in a periodic habitat. Discrete Contin. Dyn. Syst. 29, 343–366 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zhang, Y., Zhang, Q.: Dynamic behavior in a delayed stage-structured population model with stochastic fluctuation and harvesting. Nonlinear Dyn. 66, 231–245 (2011)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Tommaso Lorenzi

  2. UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, INRIA-Rocquencourt, EPI BANG, 4, pl. Jussieu, 75252, Paris cedex 05, France

    Alexander Lorz

  3. Sustainability Department, Pirelli & C. S.p.A., Via Pirelli 25, 20124, Milano, Italy

    Giorgio Restori

Authors
  1. Tommaso Lorenzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Lorz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giorgio Restori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Lorz.

Additional information

T.L. is supported by the FIRB project, RBID08PP3J. A.L. is supported by a postdoc grant from the Fondation Sciences Mathématiques de Paris.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lorenzi, T., Lorz, A. & Restori, G. Asymptotic Dynamics in Populations Structured by Sensitivity to Global Warming and Habitat Shrinking. Acta Appl Math 131, 49–67 (2014). https://doi.org/10.1007/s10440-013-9849-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-013-9849-9

Keywords

Search

Navigation